My attempt:
$$\begin{align} (\cos x- \cos 2x) – \sin 3x &= 0 \\
2\sin \frac{3x}{2}\sin \frac{x}{2} – 2 \sin \frac{3x}{2}\cos \frac{3x}{2}&=0\\
\sin\frac{3x} {2} \left(\sin\frac{x} {2} – \cos\frac{3x} {2}\right)&=0 \end{align}$$
Now either $\sin\frac{3x} {2} = 0$ or $\left(\sin\frac{x} {2} – \cos\frac{3x} {2}\right)=0$. Solving for the former,
$$\frac{3x}{2}=n\pi\rightarrow x = \frac{2n\pi}{3}$$
How can I solve $\left(\sin\frac{x} {2} – \cos\frac{3x} {2}\right)=0$? I know the elementary identities involving trigonometric ratios, but not complex numbers or calculus.
Best Answer
I think you can use the fact that $\sin(x) = \cos(\frac{\pi}{2}-x)$ to translate the equation into one involving only sines (or cosines) on both sides, and then compare directly.